Artificial aging process for aluminum alloys

ABSTRACT

Embodiments of a method for non-isothermally aging an aluminum alloy are provided. The method comprises heating an aluminum alloy at a first ramp-up rate to a maximum temperature below a precipitate solvus value, cooling the alloy at a first cooling rate sufficient to produce a maximum number of primary precipitates, cooling at a second cooling rate until a minimum temperature is reached wherein the growth rate of primary precipitates is equal to or substantially zero, and heating the alloy at a second ramp-up rate to a temperature sufficient to produce a maximum number of secondary precipitates.

TECHNICAL FIELD

Embodiments of the present invention are related to methods ofoptimizing a non-isothermal artificial aging scheme to achieve targetmaterial properties with minimum energy use and lead time.

SUMMARY

Heat treatment, in particular aging (or precipitation) hardening is animportant step to achieve the desired strength of engineering materials,such as cast aluminum alloys A356/357 or the like. Strengthening byaging hardening is applicable to alloys in which the solid solubility ofat least one alloying element decreases with decreasing temperature.Some wrought and cast aluminum alloys are age-hardenable, such as 6xxx,7xxx, 3xx, or the like. The present invention extends to all suchaluminum alloys made by various manufacturing processes including, butnot limited to forging, casting, and powder metallurgy.

Conventional heat treatment of age-hardenable aluminum alloys normallyinvolves three stages: (1) solution treatment of the products orcomponents at a relatively high temperature, for example, a temperaturejust below the melting temperature of the alloy; (2) rapid cooling (orquenching) in a cold media such as water at room-temperature or adesigned temperature; and (3) aging the materials by holding them for aperiod of time at room temperature (natural aging) or at an intermediatetemperature (artificial aging). Solution treatment serves three mainpurposes: (1) dissolution of elements that will later cause agehardening, (2) spherodization of undissolved constituents, and (3)homogenization of solute concentrations in the material.

Quenching is used to retain the solute elements in a supersaturatedsolid solution (SSS) and also to create a supersaturation of vacanciesthat enhance the diffusion and the dispersion of precipitates. Tomaximize strength of the alloy, the precipitation of all strengtheningphases should be prevented during quenching. Aging (either natural orartificial) creates a controlled dispersion of strengtheningprecipitates. FIG. 1 shows a typical heat treatment cycle of A356 castaluminum alloys. In practice, aluminum components such as cast aluminumproducts (engine blocks and cylinder heads) usually have different wallthicknesses varying from a few millimeters to a few centimeters. Due tothe conventional isothermal aging process, this leads to nonuniformitiesin temperature profile and yield strength between thin and thicksections of the aluminum product.

In the present invention, a non-isothermal aging process has beendeveloped based on precipitation strengthening and computationalthermodynamic and kinetics. The aging temperature varies with time sothat the concomitant nucleation, growth and coarsening of precipitatescan be controlled and optimized. With the non-isothermal aging scheme,the desired yield strength of aluminum alloys can be achieved withminimal time and energy. Also, uniform yield strength can be achievedacross the whole component by altering the heating/cooling scheme duringthe aging process. Higher yield strength can be realized in the improved(non-isothermal) aging process, while minimizing aging time and energyinput.

According to one embodiment of the present invention, a method fornon-isothermally aging an aluminum alloy is provided. The methodcomprises the steps of: heating an aluminum alloy at a first ramp-uprate to a maximum temperature below a precipitate solvus value, coolingthe alloy at a first cooling rate sufficient to produce a maximum numberof primary precipitates, cooling at a second cooling rate until aminimum temperature is reached wherein the growth rate of primaryprecipitates is equal to or substantially zero, and heating the alloy ata second ramp-up rate to a temperature sufficient to produce a maximumnumber of secondary precipitates.

These and additional features provided by the embodiments of the presentinvention will be more fully understood in view of the followingdetailed description, in conjunction with the drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

The following detailed description of specific embodiments of thepresent invention can be best understood when read in conjunction withthe drawings enclosed herewith. The drawing sheets include:

FIG. 1 (Prior Art) is a graphical illustration of the conventionalisothermal aging process;

FIG. 2 is a graphical illustration of the aging response of castaluminum alloys (A356/A357) aged at 170° C.;

FIG. 3 is a graphical illustration comparing the aging cycles of aconventional isothermal aging process and an embodiment of thenon-isothermal aging process according to one or more embodiments of thepresent invention; and

FIG. 4 is a comparison of the aging cycles between a conventionalisothermal aging process and two embodiments of a non-isothermal agingprocess according to one or more embodiments of the present invention.

The embodiments set forth in the drawings are illustrative in nature andnot intended to be limiting of the invention defined by the claims.Moreover, individual features of the drawings and the invention will bemore fully apparent and understood in view of the detailed description.

DETAILED DESCRIPTION

This invention is directed to achieving the maximum precipitatehardening for a given alloy (with a given amount of hardening elementsin the matrix) using minimum energy and time through a non-isothermalaging. The maximum aging hardening is obtained by producing an idealprecipitate structure comprised of uniformly distributed precipitateswhich have optimal size, shape and spacing. The size, shape and spacingis a function of aging temperature, time and concentration of hardeningelements at any given aging time and temperature.

Desirable tensile properties for cast aluminum alloys include yieldstrength and ultimate tensile strength. The ultimate tensile strength isnot an independent variable and it varies with yield strength andductility. Maximizing the yield strength is highly dependent uponprecipitate hardening. The non-isothermal aging process of thisinvention is directed to achieving this maximized yield strength withminimum energy, and minimum aging time, while also achieving a moreuniform distribution of yield strengths across the whole aluminum alloycomponent or product.

To achieve these properties, the present inventors have devised a modelwhere the age hardening process of aluminum alloys includes formation ofGuinier Preston (GP) zones, coherent and incoherent precipitates, whichis in correspondence to nucleation, growth and coarsening ofprecipitates. The contribution to the yield strength from precipitationhardening, Δσ_(ppt) is related to the microstructural (precipitate)variables:Δσ_(ppt) =f(d _(eq) ,l,f _(v) ,S,F)  (1)where d_(eq) is the average equivalent circle diameter, f_(v) is thevolume fraction of precipitates, F is the maximum interaction forcebetween an average size precipitate and dislocation, S is amicrostructural variable representing the shape and orientationrelationship of the precipitate with the matrix and dislocation line, lis the average spacing between precipitates which are acting asobstacles to dislocation motion.

The microstructural variables mentioned above are functions of agingtemperature, aging time, and solute concentrations. The contribution toyield strength from the precipitation hardening is then a function ofaging temperature, aging time, and hardening solute concentration:

$\begin{matrix}{{\Delta\sigma}_{ppt} = {A{\int_{0}^{Tc}{\int_{0}^{\infty}{\int_{0}^{c_{0}}{{f\left( {T,t,C} \right)}\ {\mathbb{d}c}\ {\mathbb{d}t}\ {\mathbb{d}T}}}}}}} & (2)\end{matrix}$where A is a constant, f(T,t,C) is the strengthening factor, C is thehardening solute concentration, and Tc is the maximum feasible agingtemperature.

For a SSS of an aluminum alloy, the age hardening process includesconcomitant nucleation, growth, and coarsening of precipitates. For agiven hardening solute concentration, concomitant nucleation, growth,and coarsening are merely sensitive to temperature and time. Thecompetition among the three processes can be manipulated to givesignificant enhancements in strength through the use of a carefullycontrolled non-isothermal aging treatment scheme, T(t), as shown in FIG.3.

In addition, the non-isothermal scheme T(t) for an aluminum alloy can beoptimized to achieve the desired yield and tensile strengths withminimum energy input and aging time. This multi-objective problem withconstraints can be defined as:

$\begin{matrix}\left\{ {{\begin{matrix}{{\underset{{({T,t})} \in \Omega}{Min}{E\left( {T,t} \right)}} = {\underset{{({T,t})} \in \Omega}{Min}{\oint{{T(t)}\mspace{11mu}{\mathbb{d}t}}}}} \\{\underset{{({T,t})} \in \Omega}{Max}{{\Delta\sigma}_{ppt}\left( {T,t,C} \right)}}\end{matrix}\Omega} = {{\left\{ {{0 < T < T_{c}};{0 < t < \infty};{0 < C < C_{0}}} \right\}{{\Delta\sigma}_{ppt}\left( {T,t,C} \right)}} \geq {\Delta\sigma}_{target}}} \right. & (3)\end{matrix}$where E(T,t) is the energy input, which is the function of temperatureand time.

In this innovative aging process, the aging scheme (cycle) is determinedby a precipitation strengthening model coupled with computationalthermodynamics and kinetics. For a SSS of an aluminum alloy, the modelsimultaneously simulates the precipitation processes includingconcomitant nucleation, growth, and coarsening. It therefore describesthe transition between shearing and bypassing of precipitates, whichcontrols the peak strength of the materials at a given agingtemperature. The model assumes that the precipitates are homogeneouslydistributed in the microstructure with a spatial size distribution andthat the dislocation line has to pass through all the obstacles(precipitates) which are encountered in the slip plane in order to causemacroscopic strain. According to dislocation strengthening theory, thestrength increase due to precipitates in the alloy can be calculated by:

$\begin{matrix}{{\Delta\sigma}_{ppt} = {\frac{M}{b}\frac{\int_{0}^{\infty}{{f\left( r_{eq} \right)}{F\left( r_{eq} \right)}{\mathbb{d}r_{eq}}}}{\int_{0}^{\infty}{{f(l)}\ {\mathbb{d}l}}}}} & (4)\end{matrix}$where Δσ_(ppt) is the strength increase due to precipitate shearing andbypassing, M is the Taylor factor; b is the Burgers vector; r_(eq) and lare precipitate equivalent circle radius (r_(eq)=0.5 d_(eq)) and spacingon the dislocation line, respectively; f(r_(eq)) is the precipitate sizedistribution; f(l) is the particle spacing distribution; and F(r_(eq))is the obstacle strength of a precipitate of radius r_(eq).

The Burgers vector, often denoted by b, is a vector that represents themagnitude and direction of the lattice distortion of dislocation in acrystal lattice. The vector b is equal to 2.86×10⁻¹⁰ m for an aluminumalloy.

Assuming solute concentrations are constant as stated above, only twolength scales (l and r_(eq)) of precipitate distribution affect thematerials strength. These two length scales are related to the agehardening process and are functions of aging temperature (T) and agingtime (t). Therefore, Eqns. (4) can be rewritten to a general form.

$\begin{matrix}{{\Delta\sigma}_{ppt} = {\frac{M}{b}{\int_{0}^{Tc}{\int_{0}^{\infty}{{f\left( {T,t} \right)}\ {\mathbb{d}t}\ {\mathbb{d}T}}}}}} & (5)\end{matrix}$

The two length scales of precipitate distribution (l and r_(eq)) can beobtained empirically from experimental measurements or by computationalthermodynamics and kinetics. In the present invention, the model istheoretically based on the fundamental nucleation and growth theories.The driving force (per mole of solute atom) for precipitation iscalculated using:

$\begin{matrix}{{\Delta\; G} = {\frac{RT}{V_{atom}}\left\lbrack {{C_{p}{\ln\left( \frac{C_{0}}{C_{eq}} \right)}} + {\left( {1 - C_{p}} \right){\ln\left( \frac{1 - C_{0}}{1 - C_{eq}} \right)}}} \right\rbrack}} & (6)\end{matrix}$where V_(atom) is the atomic volume (m³ mol⁻¹), R is the universal gasconstant (8.314 J/K mol), T is the temperature (K), C₀, C_(eq), andC_(p) are mean solute concentrations by atom percentage in matrix,equilibrium precipitate-matrix interface, and precipitates,respectively. From the driving force, a critical radius r_(eq)* isderived for the precipitates at a given matrix concentration C:

$\begin{matrix}{r_{eq}^{*} = \frac{2\gamma\; V_{atom}}{\Delta\; G}} & (7)\end{matrix}$where γ is the particle/matrix interfacial energy.

The variation of the precipitate density (number of precipitates perunit volume) is given by the nucleation rate. The evolution of the meanprecipitate size (radius) is given by the combination of the growth ofexisting precipitates and the addition of new precipitates at thecritical nucleation radius r_(eq)*. The nucleation rate is calculatedusing a standard Becker-Döring law:

$\begin{matrix}{\left. \frac{\mathbb{d}N}{\mathbb{d}t} \right|_{nucleation} = {N_{0}Z\;\beta*{\exp\left( {- \frac{4\pi\; r_{0}^{2}\gamma}{3{RT}\;{\ln^{2}\left( {C/C_{eq}} \right)}}} \right)}{\exp\left( {- \frac{1}{2\beta*{Zt}}} \right)}}} & (8)\end{matrix}$where N is the precipitate density (number of precipitates per unitvolume), N₀ is the number of atoms per unit volume (=1/V_(atom)), Z isZeldovich's factor (≈ 1/20). The evolution of the precipitate size iscalculated by:

$\begin{matrix}{\frac{\mathbb{d}r_{eq}}{\mathbb{d}t} = {{\frac{D}{r_{eq}}\frac{C - {C_{eq}{\exp\left( {r_{0}/r_{eq}} \right)}}}{1 - {C_{eq}{\exp\left( {r_{0}/r_{eq}} \right)}}}} + {\frac{1}{N}\frac{\mathbb{d}N}{\mathbb{d}t}\left( {{\alpha\;\frac{r_{0}}{\ln\left( {C/C_{eq}} \right)}} - r_{eq}} \right)}}} & (9)\end{matrix}$where D is the diffusion coefficient of solute atom in solvent.

In the late stages of precipitation, the precipitates continue growingand coarsening, while the nucleation rate decreases significantly due tothe desaturation of solid solution. When the mean precipitate size ismuch larger than the critical radius, it is valid to consider growthonly. When the mean radius and the critical radius are equal, theconditions for the standard Lifshitz-Slyozov-Wagner (LSW) law arefulfilled. Under the LSW law, the radius of a growing particle is afunction of t^(1/3) (t is the time). The precipitate radius can becalculated by:

$\begin{matrix}{{r_{eq}^{3} - r_{o}^{3}} = {\frac{8}{9}\frac{D\; C_{o}\gamma\; V_{atom}^{2}t}{RT}}} & (10)\end{matrix}$

Several assumptions are made in calculating the particle spacing alongthe dislocation line. First, a steady state number of precipitates alongthe moving dislocation line is assumed, following Friedel's statisticsfor low obstacle strengths. After assuming a steady state number ofprecipitates, the precipitate spacing is then given by the calculationof the dislocation curvature under the applied resolved shear stress, τon the slip plane:

$\begin{matrix}{l = \left( {\frac{4\pi}{3f_{v}}\frac{\overset{\_}{r_{eq}^{2}}\Gamma}{b\;\tau}} \right)^{1/3}} & (11)\end{matrix}$where f_(v) is the volume fraction of precipitates and r_(eq) is theaverage radius of precipitates. Γ is the line tension (=βμb², where β isa parameter close to ½).

The volume fraction of precipitates (f_(v)) can be determinedexperimentally by Transmission Electron Microscopy (TEM) or theHierarchical Hybrid Control (HHC) model. In the HHC model, the volumefraction of precipitates can be calculated:

$\begin{matrix}{f_{v} = {\frac{2\pi\; r_{eq}^{3}}{\alpha}A_{0}N_{0}Z\;\beta^{*}{\exp\left( \frac{{- \Delta}\; G^{*}}{RT} \right)}t}} & (12)\end{matrix}$where α is the aspect ratio of precipitates, A₀ is the Avogadro number,ΔG* is the critical activation energy for precipitation, the parameterof β* is obtained byβ*=4π(r _(eq)*)DC ₀ /a ⁴  (13)where a is the lattice parameter of precipitate.

In computational thermodynamics approaches, a commercially availablealuminum database, for instance Pandat®, is employed to calculateprecipitate equilibriums, such as β phase in Al—Si—Mg alloy and θ phasein Al—Si—Mg—Cu alloy. The equilibrium phase fractions, or the atomic %solute in the hardening phases are parameterized from computationalthermodynamics calculations. The equilibrium phase fractions aredependent upon temperature and solute concentration, but independent ofaging time (f_(i) ^(eq) (T,C)).

Many metastable precipitate phases, such as β″, β′ in Al—Si—Mg alloy andθ′ in Al—Si—Mg—Cu alloy are absent from the existing computationalthermodynamics database. The computational thermodynamics calculationsalone cannot deliver the values of metastable phase fractions. In thiscase, the density-functional based first-principles methods are adoptedto produce some properties such as energetics, which are needed bycomputational thermodynamics. Density functional theory (DFT) is aquantum mechanical theory commonly used in physics and chemistry toinvestigate the ground state of many-body systems, in particular atoms,molecules and the condensed phases. The main idea of DFT is to describean interacting system of fermions via its density and not via itsmany-body wave function. First-principles methods, also based onquantum-mechanical electronic structure theory of solids, produceproperties such as energetics without reference to any experimentaldata. The free energies of metastable phases can be described by asimple linear functional form:ΔG _(i)(T)=c ₁ +c ₂ T  (14)where c₁ and c₂ are coefficients. c₁ is equivalent to enthalpies offormation of metastable phases at absolute zero temperature (T=0 K). Byreplacing the unknown parameter c₁ in Eqn. 14 with the formationenthalpy at T=0 K from first-principles, the free energy can berewritten asΔG _(i)(T)=ΔH _(i)(T=0K)+c ₂ T  (15)The other unknown parameter c₂ can then be determined simply by fittingthe free energies of liquid and solid to be equal at the melting point.

After calculating the strength increase due to precipitation hardening(Δσ_(ppt)), the yield strength of aluminum alloys can be simplycalculated by adding it to the intrinsic strength (σ_(i)) and thesolid-solution strength of the material:σ_(ys)=σ_(i)+σ_(ss)+Δσ_(ppt)  (16)

The solid solution contribution to the yield strength is calculated as:σ_(ss) =KC _(GP/ss) ^(2/3)  (17)where K is a constant and C_(GP/ss) is the concentration ofstrengthening solute that is not in the precipitates. The intrinsicstrength (σ_(i)) includes various strengthening effects such asgrain/cell boundaries, the eutectic particles (in cast aluminum alloys),the aluminum matrix, and solid-solution strengthening due to alloyingelements other than elements in precipitates.

This aging profile may be customized for various alloys with varyingtemperature profiles. In one embodiment, the non-isothermal agingprocess may include the step of heating an aluminum alloy at a firstramp-up rate to a maximum temperature below the precipitate solvus. Byselecting a maximum temperature just below the precipitate solvus, thenumber of stable primary precipitate nuclei is maximized. As usedherein, the precipitate solvus is the limit of solubility for ahomogeneous solid solution before it will be degraded through melting,etc. The precipitate solvus temperature can be either measured orcalculated. In an A356 alloy (7% Si and 0.4% Mg), the solvus temperaturefor the β″ precipitates is about 280° C. The first ramp-up rate may bethe maximum possible heating rate. In one exemplary embodiment, thefirst ramp-up rate may be up to about 100° C./s.

After the maximum temperature is reached, the alloy may be cooled at afirst cooling rate sufficient to produce a maximum number of primaryprecipitates. The primary precipitates may be arranged in a homogenousvolumetric distribution in simple or complex shaped components. Complexshaped components may include but are not limited to engine blocks orcylinder heads. Primary precipitates are typically those grown in thealloy in the underaged or peak aged stages as shown in FIG. 2. The firstcooling rate may be obtained utilizing various equations familiar to oneof ordinary skill in the art. In one embodiment, the first cooling ratemay be obtained by optimizing precipitation growth rate

$\frac{\mathbb{d}r_{eq}}{\mathbb{d}t}$and nucleation rate

$\frac{\mathbb{d}N}{\mathbb{d}t}$using equations such as 8 and 9 shown below:

${{\frac{\mathbb{d}N}{\mathbb{d}t}❘_{nucleation}} = {N_{0}Z\;\beta^{*}{\exp\left( {- \frac{4\pi\; r_{0}^{2}\gamma}{3{RT}\;{\ln^{2}\left( {C/C_{eq}} \right)}}} \right)}{\exp\left( {- \frac{1}{2\beta^{*}{Zt}}} \right)}}}\mspace{11mu}$and$\frac{\mathbb{d}r_{eq}}{\mathbb{d}t} = {{\frac{D}{r_{eq}}\frac{C - {C_{eq}{\exp\left( {r_{0}/r_{eq}} \right)}}}{1 - {C_{eq}{\exp\left( {r_{0}/r_{eq}} \right)}}}} + {\frac{1}{N}\frac{\mathbb{d}N}{\mathbb{d}t}\left( {{\alpha\frac{r_{0}}{\ln\left( {C/C_{eq}} \right)}} - r_{eq}} \right)}}$

The optimization is characterized by the maximization of

$\frac{\mathbb{d}N}{\mathbb{d}t}$and the minimization of

$\frac{\mathbb{d}r_{eq}}{\mathbb{d}t}.$The optimization of these variables and equations may be conducted viaan optimization algorithm familiar to one of ordinary skill in the art,for example, a computerized algorithm or iterative algorithm.

Subsequently, the alloy is cooled at a more rapid second cooling rateuntil a minimum temperature is reached wherein the growth rate ofexisting precipitates is at or close to zero. The second cooling rate istypically designed to lower the temperature as quickly as possiblewithin practical equipment limits. Many methods of calculating thesecond cooling rate are contemplated herein. In one embodiment, minimumtemperature may be obtained by via equations 8 and 9. At the minimumtemperature, the precipitation growth rate

$\frac{\mathbb{d}r_{eq}}{\mathbb{d}t}$is at or approaching zero, thus

$\frac{\mathbb{d}r_{eq}}{\mathbb{d}t}$in equations 8 and 9 is set to zero and the minimum temperature may besolved.

After the minimum temperature is achieved, the alloy is heated at asecond ramp-up rate to a temperature sufficient to produce a maximumnumber of homogeneously distributed secondary precipitates. Secondaryprecipitates may occur in the overaged stage as shown in FIG. 2. Thesecond ramp-up rate is obtained by optimizing the precipitation growthrate and the nucleation rate using equations such as 8 and 9, whileadjusting for the composition change due to the formation of the primaryprecipitates. The optimization is characterized by the maximization of

$\frac{\mathbb{d}N}{\mathbb{d}t}\mspace{11mu}{and}\mspace{11mu}{\frac{\mathbb{d}r_{eq}}{\mathbb{d}t}.}$The second ramp-up rate is configured to minimize the growth rate andnucleate as many secondary precipitates as possible.

Additionally, equations 10 through 15 may be optimized to ensure thatthe final temperature and second ramp-up rate are controlled to yieldthe highest number density of secondary precipitates. The finaltemperature and second ramp-up rate are further optimized for the energyminimization and target strength, wherein the target strength constrainthelps prevent coarsening the primary precipitates while producing thesecondary precipitates. The strength, which may be calculated withequation 16, depends on the number and sizes of precipitate particles,in addition to how closely spaced the particles are.

As shown in Exp. 1 of FIG. 4, embodiments of the present invention mayalso be directed to a process of achieving a target strength with lowerenergy using a single step process to optimize primary precipitates.This may be achieved by controlling the cooling rate alone, withoututilizing a secondary precipitate control step.

Using equation 4 above, the maximum tensile strength increase due toprecipitation Δσ_(ppt) may be calculated. In one embodiment, the agingprocess yields a tensile strength of about 250 to about 300 MPa, andrequires from about 750 to about 800° C.*hr (energy index) in energyinput over 5 hours.

The energy index is derived as follows. Assuming that the surface areaof the furnace is A (m²) and the wall thickness of the furnace is L (m).The heat flux of energy lost (input) through heat conduction at a giventime is:

$H = {{k \cdot \frac{A}{L}}\left( {{T(t)} - T_{air}} \right)}$where k is the thermal conductivity of the wall material in the furnace.T(t) and T_(air) are temperatures of furnace and air, respectively.

For a period of time (t), the energy loss (input) is then:

$Q = {{\int_{0}^{t}{H{\mathbb{d}t}}} = {{k\frac{A}{L}{\int_{o}^{t}{\left( {{T(t)} - T_{air}} \right){\mathbb{d}t}}}} = {k\frac{A}{L}Q_{I}}}}$Q_(I) = ∫_(o)^(t)(T(t) − T_(air))𝕕twhere Q_(I) is the energy index (unit: ° C.*hr), which is theintegration of aging temperature over the entire aging time.

Referring to FIG. 4 and Table 1 below, the non-isothermal aging Exp. 1and 2 were compared with a conventional isothermal aging cycle. Forcomparison, a conventional isothermal aging cycle is assumed at 170° C.(or 443° K) for 5.4 hrs. The total aging time in non-isothermal Exp. 1is 5 hrs. In comparison with the conventional isothermal aging (170° C.for 5.4 hrs), the non-isothermal aging Exp. 2 provides reduced energyinput (saving ˜15%), reduced aging time, while achieving increased yieldstrength (increased ˜10%).

TABLE 1 Temperature Aging time Energy Input Index Yield strength (MPa)Aging cycle (° C.) (hrs) (° C. × hr) Measured Predicted Conventional 1705.4 918 252 249 isothermal aging Non-isothermal vary 5 852 204 211 agingExp 1 Non-isothermal vary 5 792 278 275 aging Exp 2

For the purposes of describing and defining the present invention it isnoted that the terms “substantially” and “about” are utilized herein torepresent the inherent degree of uncertainty that may be attributed toany quantitative comparison, value, measurement, or otherrepresentation. These terms are also utilized herein to represent thedegree by which a quantitative representation may vary from a statedreference without resulting in a change in the basic function of thesubject matter at issue.

Having described the invention in detail and by reference to specificembodiments thereof, it will be apparent that modifications andvariations are possible without departing from the scope of theinvention defined in the appended claims. More specifically, althoughsome aspects of the present invention are identified herein as preferredor particularly advantageous, it is contemplated that the presentinvention is not necessarily limited to these preferred aspects of theinvention.

1. A method for non-isothermally aging an aluminum alloy comprising:heating an aluminum alloy at a first ramp-up rate to a maximumtemperature below a precipitate solvus; when the aluminum alloy reachesthe maximum temperature, cooling the alloy at a first cooling ratesufficient to produce a maximum number of primary precipitates whereinthe first cooling rate is obtained by optimizing a precipitation growthrate $\frac{\mathbb{d}r_{eq}}{\mathbb{d}t}$ and a nucleation rate$\frac{\mathbb{d}N}{\mathbb{d}t}$ using the following two equations:${{\frac{\mathbb{d}N}{\mathbb{d}t}❘_{nucleation}} = {N_{0}Z\;\beta^{*}{\exp\left( {- \frac{4\pi\; r_{0}^{2}\gamma}{3{RT}\;{\ln^{2}\left( {C/C_{eq}} \right)}}} \right)}{\exp\left( {- \frac{1}{2\beta^{*}{Zt}}} \right)}}}\mspace{11mu}$and${\frac{\mathbb{d}r_{eq}}{\mathbb{d}t} = {{\frac{D}{r_{eq}}\frac{C - {C_{eq}{\exp\left( {r_{0}/r_{eq}} \right)}}}{1 - {C_{eq}{\exp\left( {r_{0}/r_{eq}} \right)}}}} + {\frac{1}{N}\frac{\mathbb{d}N}{\mathbb{d}t}\left( {{\alpha\frac{r_{0}}{\ln\left( {C/C_{eq}} \right)}} - r_{eq}} \right)}}},$where N is the precipitate density number (number of precipitates perunit volume), N₀ is the number of atoms per unit volume (=1/V_(atom)), Zis Zeldovich's factor, $\frac{\mathbb{d}r_{eq}}{\mathbb{d}t}$ is theprecipitation growth rate, D is the diffusion constant, r_(eq) is theprecipitate radius (also called precipitate size), r₀ is the value of$\frac{2\gamma\; V_{atom}}{R\; T},$ C₀ is the mean solute concentrationby atom percentage in the alloy matrix, C_(eq) is the mean soluteconcentration by atom percentage in equilibrium precipitate-matrixinterface, and α is the aspect ratio of precipitates, wherein theoptimization is characterized by the maximization of$\frac{\mathbb{d}N}{\mathbb{d}t}$ and the minimization of$\frac{\mathbb{d}r_{eq}}{\mathbb{d}t};$ after cooling the alloy at thefirst cooling rate, cooling the alloy at a second cooling rate until aminimum temperature is reached wherein the growth rate of primaryprecipitates is equal to or substantially zero, the second cooling ratebeing higher than the first cooling rate; and when the minimumtemperature is reached, heating the alloy at a second ramp-up rate to atemperature sufficient to produce a maximum number of secondaryprecipitates; the first ramp-up rate, the first cooling rate, the secondcooling rate, and the second ramp-up rate causing non-isothermal agingin which an aging temperature varies continuously with time.
 2. Themethod of claim 1 wherein the primary precipitates and the secondaryprecipitates are homogeneously distributed.
 3. The method of claim 1wherein the alloy is present in a complex shaped component.
 4. Themethod of claim 3 wherein the complex shaped component is an engineblock or cylinder head.
 5. The method of claim 1 wherein the firstramp-up rate is the maximum achievable heating rate.
 6. The method ofclaim 1 wherein the first ramp-up rate is up to about 100° C./s.
 7. Themethod of claim 1 wherein the second cooling rate is the maximumachievable cooling rate.
 8. The method of claim 1 wherein the minimumtemperature is obtained by the equation${\frac{\mathbb{d}r_{eq}}{\mathbb{d}t} = {{\frac{D}{r_{eq}}\frac{C - {C_{eq}{\exp\left( {r_{0}/r_{eq}} \right)}}}{1 - {C_{eq}{\exp\left( {r_{0}/r_{eq}} \right)}}}} + {\frac{1}{N}\frac{\mathbb{d}N}{\mathbb{d}t}\left( {{\alpha\frac{r_{0}}{\ln\left( {C/C_{eq}} \right)}} - r_{eq}} \right)}}},{where}$$\frac{\mathbb{d}r_{eq}}{\mathbb{d}t}$ is the precipitation growth rate,D is the diffusion constant, r_(eq) is the precipitate radius (alsocalled precipitate size), r₀ is the value of$\frac{2\gamma\; V_{atom}}{R\; T},$ C₀ is the mean solute concentrationby atom percentage in the alloy matrix, C_(eq) is the mean soluteconcentration by atom percentage in equilibrium precipitate-matrixinterface, and α is the aspect ratio of precipitates, wherein$\frac{\mathbb{d}r_{eq}}{\mathbb{d}t} = 0$ at the minimum temperature.9. The method of claim 1 wherein the second ramp-up rate is obtained byoptimizing the precipitation growth rate and the nucleation rate usingthe following two equations:$\left. \frac{\mathbb{d}N}{\mathbb{d}t} \right|_{nucleation} = {N_{0}Z\;\beta^{*}{\exp\left( {- \frac{4\pi\; r_{0}^{2}\gamma}{3R\; T\;{\ln^{2}\left( {C/C_{eq}} \right)}}} \right)}{\exp\left( {- \frac{1}{2\beta^{*}Z\; t}} \right)}}$and${\frac{\mathbb{d}r_{eq}}{\mathbb{d}t} = {{\frac{D}{r_{eq}}\frac{C - {C_{eq}{\exp\left( {r_{0}/r_{eq}} \right)}}}{1 - {C_{eq}{\exp\left( {r_{0}/r_{eq}} \right)}}}} + {\frac{1}{N}\frac{\mathbb{d}N}{\mathbb{d}t}\left( {{\alpha\frac{r_{0}}{\ln\left( {C/C_{eq}} \right)}} - r_{eq}} \right)}}},$where N is the precipitate density (number of precipitates per unitvolume), N₀ is the number of atoms per unit volume (=1/V_(atom)), Z isZeldovich's factor, $\frac{\mathbb{d}r_{eq}}{\mathbb{d}t}$ is theprecipitation growth rate, D is the diffusion constant, r_(eq) is theprecipitate radius (also called precipitate size), r₀ is the value of$\frac{2\gamma\; V_{atom}}{R\; T},$ C₀ is the mean solute concentrationby atom percentage in the alloy matrix, C_(eq) is the mean soluteconcentration by atom percentage in equilibrium precipitate-matrixinterface, and α is the aspect ratio of precipitates, wherein theoptimization is characterized by the maximization of$\frac{\mathbb{d}N}{\mathbb{d}t}\mspace{14mu}{and}\mspace{14mu}{\frac{\mathbb{d}r_{eq}}{\mathbb{d}t}.}$10. The method of claim 1 wherein the aging achieves a maximum tensilestrength increase due to precipitation Δσ_(ppt) according to theequation${\Delta\;\sigma_{ppt}} = {\frac{M}{b}\frac{\int_{0}^{\infty}{{f\left( r_{eq} \right)}{F\left( r_{eq} \right)}{\mathbb{d}r_{eq}}}}{\int_{0}^{\infty}{{f(l)}{\mathbb{d}l}}}}$where M is the Taylor factor, b is the Burgers vector, r_(eq) is theprecipitate radius (also called precipitate size), l is the spacing onthe dislocation line, f(r_(eq)) is the precipitate size distribution,f(l) is the particle spacing distribution, and F(r_(eq)) is the obstaclestrength of a precipitate of radius r_(eq).
 11. The method of claim 10wherein l is equal to${l = \left( {\frac{4\pi}{3f_{v}}\frac{\overset{\_}{r_{eq}^{2}}\Gamma}{b\;\tau}} \right)^{1/3}},$where f_(v) is the volume fraction of precipitates and r_(eq) is theaverage radius of precipitates, Γ is the line tension.
 12. The method ofclaim 11 where$f_{v} = {\frac{2\pi\; r_{eq}^{3}}{\alpha}A_{0}N_{0}Z\;\beta^{*}{\exp\left( \frac{{- \Delta}\; G^{*}}{R\; T} \right)}t}$and β*=4π(r*_(eq))²DC₀/a⁴.
 13. The method of claim 11 wherein the linetension is βμb², where β is approximately ½.
 14. The method of claim 10wherein r_(eq) is defined by the equation${r_{eq}^{3} - r_{o}^{3}} = {\frac{8}{9}\frac{D\; C_{o}\gamma\; V_{atom}^{2}t}{R\; T}}$when the mean precipitate size is much larger than critical radiusr_(eq*).
 15. A method for producing an aluminum alloy comprising:solution treating the alloy at temperatures below the melting point ofthe alloy; quenching the solution treated alloy; and aging the quenchedalloy according to the method of claim 1.